How does a DSLR help me to get better pictures than a good point-and-shoot camera?
Different lenses, mainly. Probably a larger sensor than what you have now. Maybe some better low light performance. Maybe a lower aperture setting because of the lense, like 2.8 or 1.8, etc. The DSLR bodies are also just a little bigger, and consider the knobs, wheels, and buttons, as just part of a tool. They give you easier access to change settings on the fly to suit the situation. It's like a $1000 miter saw versus a $200 one. They'll both cut wood, but one's gonna have more features, maybe make it easier, smoother cuts, etc.
That being said, Matjam already said what I would say, kind of. "The best camera is the one you have with you." My cell phone is the best camera I have because it's always with me. When I travel, I bring my Nikon P7000 point and shoot. I used to shoot weddings professionally, so I have a D700, a D300, a D80, and SB 900/600 strobes, and tons of 2.8 zoom glass. But it's heavy, and burdensome, so it's not portable to take out to parks with my wife and daughter, or on trips. So my point and shoot and phone become better options. I'll bring my D80 with a 50mm prime with us quite a bit, but even then I'd rather just leave it at home most of the time.
But if you do love photography, and do want to get more into it, the capabilities of a DSLR is a good investment and you'd love it. As a guy with lots of camera bodies, I won't talk you out of it. If you have the money, go for it. But it's nice to keep a little point and shoot around too. A DSLR won't naturally take "better" pictures, it will just give you a better tool to do what you are already doing.
It's better to have non-DSLR pictures than no pictures at all. :)